TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000–000
S 0002-9947(XX)0000-0
CROSSINGS AND NESTINGS OF MATCHINGS AND
PARTITIONS
WILLIAM Y.C. CHEN, EVA Y.P. DENG, ROSENA R.X. DU, RICHARD P. STANLEY,
AND CATHERINE H. YAN
Abstract. We present results on the enumeration of crossings and nestings
for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and
maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers
and the nesting numbers are distributed symmetrically over all partitions of
[n], as well as over all matchings on [2n]. As a corollary, the number of knoncrossing partitions is equal to the number of k-nonnesting partitions. The
same is also true for matchings. An application is given to the enumeration of
matchings with no k-crossing (or with no k-nesting).
1. Introduction
A (complete) matching on [2n] = {1, 2, . . . , 2n} is a partition of [2n] of type
(2, 2, . . . , 2). It can be represented by listing its n blocks, as {(i1 , j1 ), (i2 , j2 ), . . . ,
(in , jn )}, where ir < jr for 1 ≤ r ≤ n. Two blocks (also called arcs) (ir , jr ) and
(is , js ) form a crossing if ir < is < jr < js ; they form a nesting if ir < is < js < jr .
It is well-known that the number of matchings on [2n] with no crossings (or with
no nestings) is given by the n-th Catalan number
2n
1
.
Cn =
n+1 n
See [25, Exercise 6.19] for many combinatorial interpretations of Catalan numbers,
where item (o) is for noncrossing matchings, and item (ww) can be viewed as
nonnesting matchings, in which the blocks of the matching are the columns of the
standard Young tableaux of shape (n, n). Nonnesting matchings are also one of the
items of [26].
Let k ≥ 2 be an integer. A k-crossing of a matching M is a set of k arcs (ir1 , jr1 ),
(ir2 , jr2 ), . . . , (irk , jrk ) of M such that ir1 < ir2 < · · · < irk < jr1 < jr2 < · · · < jrk .
A matching without any k-crossing is a k-noncrossing matching. Similarly, a knesting is a set of k arcs (ir1 , jr1 ), (ir2 , jr2 ), . . . , (irk , jrk ) of M such that ir1 < ir2
Received by the editors April 19, 2005 and, in revised form, November 9, 2005.
2000 Mathematics Subject Classification. Primary 05A18; Secondary 05E10, 05A15.
Key words and phrases. Crossing, nesting, partition, vacillating tableau.
The first author was supported by the 973 Project on Mathematical Mechanization, the National Science Foundation, the Ministry of Education and the Ministry of Science and Technology
of China.
The fourth author was supported in part by NSF grant #DMS-9988459.
The fifth author was supported in part by NSF grant #DMS-0245526 and a Sloan Fellowship.
c
1997
American Mathematical Society
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W. CHEN, E. DENG, R. DU, R. STANLEY, AND C. YAN
< · · · < irk < jrk < · · · < jr2 < jr1 . A matching without any k-nesting is a
k-nonnesting matching.
Enumeration on crossings/nestings of matchings has been studied for the cases
k = 2 and k = 3. For k = 2, in addition to the above results on Catalan numbers,
the distribution of the number of 2-crossings has been studied by Touchard [29], and
later more explicitly by Riordan [21], who gave a generating function. M. de SainteCatherine [8] proved that 2-crossings and 2-nestings are identically distributed over
all matchings of [2n], i.e., the number of matchings with r 2-crossings is equal to
the number of matchings with r 2-nestings.
The enumeration of 3-nonnesting matchings was first studied in [11] by Gouyou
-Beauschamps, where he gave a bijection between involutions with no decreasing sequence of length 6 and pairs of noncrossing Dyck left factors by a recursive construction. His bijection is essentially a correspondence between 3-nonnesting matchings
and pairs of noncrossing Dyck paths, where a matching can also be considered as a
fixed-point-free involution. We observed that the number of 3-noncrossing matchings also equals the number of pairs of noncrossing Dyck paths, and a one-to-one
correspondence between 3-noncrossing matchings and pairs of noncrossing Dyck
paths can be built recursively.
In this paper, we extend the above results. Let cr(M ) be maximal i such that M
has an i-crossing, and ne(M ) the maximal j such that M has a j-nesting. Denoted
by fn (i, j) the number of matchings M on [2n] with cr(M ) = i and ne(M ) = j. We
shall prove that fn (i, j) = fn (j, i). As a corollary, the number of matchings on [2n]
with cr(M ) = k equals the number of matchings M on [2n] with ne(M ) = k.
Our construction applies to a more general structure, viz., partitions of a set.
Given a partition P of [n], denoted by P ∈ Πn , we represent P by a graph on
the vertex set [n] whose edge set consists of arcs connecting the elements of each
block in numerical order. Such an edge set is called the standard representation of the partition P . For example, the standard representation of 1457-26-3
is {(1, 4), (4, 5), (5, 7), (2, 6)}. Here we always write an arc e as a pair (i, j) with
i < j, and say that i is the lefthand endpoint of e and j is the righthand endpoint
of e.
Let k ≥ 2 and P ∈ Πn . Define a k-crossing of P as a k-subset (i1 , j1 ), (i2 , j2 ), . . . ,
(ik , jk ) of the arcs in the standard representation of P such that i1 < i2 < · · · <
ik < j1 < j2 < · · · < jk . Let cr(P ) be the maximal k such that P has a k-crossing.
Similarly, define a k-nesting of P as a k-subset (i1 , j1 ), (i2 , j2 ), . . . , (ik , jk ) of the
set of arcs in the standard representation of P such that i1 < i2 < · · · < ik < jk <
· · · < j2 < j1 , and ne(P ) the maximal j such that P has a j-nesting. Note that
when restricted to complete matchings, these definitions agree with the ones given
before.
Let gn (i, j) be the number of partitions P of [n] with cr(P ) = i and ne(P ) = j.
We shall prove that gn (i, j) = gn (j, i), for all i, j and n. In fact, our result is much
stronger. We present a generalization which implies the symmetric distribution of
cr(P ) and ne(P ) over all partitions in Πn , as well over all complete matchings on
[2n].
To state the main result, we need some notation. Given P ∈ Πn , define
min(P ) = {minimal block elements of P },
max(P ) = {maximal block elements of P }.
CROSSINGS AND NESTINGS
3
For example, for P = 135-26-4, min(P ) = {1, 2, 4} and max(P ) = {4, 5, 6}. The
pair (min(P ), max(P )) encodes some useful information about the partition P . For
example, the number of blocks of P is | min(P )| = | max(P )|; number of singleton
blocks is | min(P ) ∩ max(P )|; P is a (partial) matching if and only if min(P ) ∪
max(P ) = [n], and P is a complete matching if in addition, min(P ) ∩ max(P ) = ∅.
Fix S, T ⊆ [n] with |S| = |T |. Let Pn (S, T ) be the set {P ∈ Πn : min(P ) =
S, max(P ) = T }, and fn,S,T (i, j) be the cardinality of the set {P ∈ Pn (S, T ) :
cr(P ) = i, ne(P ) = j}.
Theorem 1.1.
(1.1)
fn,S,T (i, j) = fn,S,T (j, i).
In other words,
(1.2)
X
P ∈Pn (S,T )
X
xcr(P ) y ne(P ) =
xne(P ) y cr(P ) .
P ∈Pn (S,T )
That is, the statistics cr(P ) and ne(P ) have a symmetric joint distribution over
each set Pn (S, T ). Summing over all pairs (S, T ) in (1.1), we get
(1.3)
gn (i, j) = gn (j, i).
We say that a partition P is k-noncrossing if cr(P ) < k. It is k-nonnesting if
ne(P ) < k. Let NCNk,l (n) be the number of partitions of [n] that are k-noncrossing
and l-nonnesting. Summing over 1 ≤ i < k and 1 ≤ j < l in (1.3), we get the
following corollary.
Corollary 1.2. NCNk,l (n) = NCNl,k (n).
Letting l > n, Corollary 1.2 becomes the following result.
Corollary 1.3. NCk (n) = NNk (n), where NCk (n) is the number of k-noncrossing
partitions of [n], and NNk (n) is the number of k-nonnesting partitions of [n]. Theorem 1.1 also applies to complete matchings. A partition P of [2n] is a
complete matching if and only if | min(P )| = | max(P )| = n and min(P )∩max(P ) =
∅. (It follows that min(P ) ∪ max(P ) = [2n].) Restricting Theorem 1.1 to disjoint
pairs (S, T ) of [2n] with |S| = |T | = n, we get the following result on the crossing
and nesting number of complete matchings.
Corollary 1.4. Let M be a matching on [2n].
1. The statistics cr(M ) and ne(M ) have a symmetric joint distribution over P2n (S, T ),
where |S| = |T | = n, and S, T are disjoint.
2. fn (i, j) = fn (j, i) where fn (i, j) is the number of matchings on [2n] with cr(M ) =
i and ne(M ) = j.
3. The number of matchings on [2n] that are k-noncrossing and l-nonnesting is
equal to the number of matchings on [2n] that are l-noncrossing and k-nonnesting.
4. The number of k-noncrossing matchings on [2n] is equal to the number of knonnesting matchings on [2n].
The paper is arranged as follows. In Section 2 we introduce the concept of
vacillating tableau of general shape, and give a bijective proof for the number of
vacillating tableaux of shape λ and length 2n. In Section 3 we apply the bijection
4
W. CHEN, E. DENG, R. DU, R. STANLEY, AND C. YAN
of Section 2 to vacillating tableaux of empty shape, and characterize crossings and
nestings of a partition by the corresponding vacillating tableau. The involution
on the set of vacillating tableaux defined by taking the conjugate to each shape
leads to an involution on partitions which exchanges the statistics cr(P ) and ne(P )
while preserves min(P ) and max(P ), thus proving Theorem 1.1. Then we modify
the bijection between partitions and vacillating tableaux by taking isolated points
into consideration, and give an analogous result on the enhanced crossing number
and nesting number. This is the content of Section 4. Finally in Section 5 we
restrict our bijection to the set of complete matchings and oscillating tableaux,
and study the enumeration of k-noncrossing matchings. In particular, we construct
bijections from k-noncrossing matchings for k = 2 or 3 to Dyck paths and pairs of
noncrossing Dyck paths, respectively, and present the generating function for the
number of k-noncrossing matchings.
2. A Bijection between Set Partitions and Vacillating Tableaux
Let Y be Young’s lattice, that is, the set of all partitions of all integers n ∈ N
ordered component-wise, i.e.,
P (µ1 , µ2 , . . . ) ≤ (λ1 , λ2 , . . . , ) if µi ≤ λi for all i. We
write λ ` k or |λ| = k if
λi = k. A vacillating tableau is a walk on the Hasse
diagram of Young’s lattice subject to certain conditions. The main tool in our
proof of Theorem 1.1 is a bijection between the set of set partitions and the set of
vacillating tableaux of empty shape ∅.
Definition 2.1. A vacillating tableau Vλ2n of shape λ and length 2n is a sequence
λ0 , λ1 , . . . , λ2n of integer partitions such that (i) λ0 = ∅, and λ2n = λ, (ii) λ2i+1
is obtained from λ2i by doing nothing (i.e., λ2i+1 = λ2i ) or deleting a square, and
(iii) λ2i is obtained from λ2i−1 by doing nothing or adding a square.
In other words, a vacillating tableau of shape λ is a walk on the Hasse diagram
of Young’s lattice from ∅ to λ where each step consists of either (i) doing nothing
twice, (ii) do nothing then adding a square, (iii) removing a square then doing
nothing, or (iv) removing a square and then adding a square. Note that if the
length is larger than 0, λ1 = ∅. If the vacillating tableau is of empty shape, then
λ2n−1 = ∅ as well.
Example 2.2. Abbreviate λ = (λ1 , λ2 , . . . ) by λ1 λ2 · · · . There are 5 vacillating
tableaux of shape ∅ and length 6. They are
(2.1)
λ0
∅
∅
∅
∅
∅
λ1
∅
∅
∅
∅
∅
λ2
∅
∅
1
1
1
λ3
∅
∅
∅
∅
1
λ4
∅
1
1
∅
1
λ5
∅
∅
∅
∅
∅
λ6
∅
∅
∅
∅
∅
Example 2.3. An example of a vacillating tableau of shape 11 and length 10 is
given by
∅, ∅, 1, 1, 1, 1, 2, 2, 21, 11, 11.
Theorem 2.4. (i) Let gλ (n) be the number of vacillating tableaux of shape λ ` k
and length 2n. By a standard Young tableau (SYT) of shape λ, we mean an array
T of shape λ whose entries are distinct positive integers that increase in every row
CROSSINGS AND NESTINGS
5
and column. The content of T is the set of positive integers that appear in it. (We
don’t require that content(λ) = [k], where λ ` k.) We then have
gλ (n) = B(n, k)f λ ,
where f λ is the number of SYT’s of shape λ and content [k], and B(n, k) is the
number of partitions of [n] with k blocks distinguished.
(ii) The exponential generating function of B(n, k) is given by
(2.2)
X
n≥0
B(n, k)
xn
1
= (ex − 1)k exp(ex − 1).
n!
k!
To prove Theorem 2.4, we construct a bijection between the set Vλ2n of vacillating
tableaux of shape λ and length 2n, and pairs (P, T ), where P is a partition of [n],
and T is an SYT of shape λ such that content(T ) ⊆ max(P ). In the next section
we apply this bijection to vacillating tableaux of empty shape, and related it to the
enumeration of crossing and nesting numbers of a partition. In the following we
shall assume familiarity with the RSK algorithm, and use row-insertion P ←− k
as the basic operation of the RSK algorithm. For the notation, as well as some
basic properties of the RSK algorithm, see e.g. [25, Chapter 7]. In general we
shall apply the RSK algorithm to a sequence w of distinct integers, denoted by
RSK
w 7−→ (A(w), B(w)), where A(w) is the (row)-insertion tableau and B(w) the
recording tableau. The shape of the SYT’s A(w) and B(w) is also called the shape
of the sequence w.
The Bijection ψ from vacillating tableaux to pairs (P, T ). Given a vacillating tableau V = (∅ = λ0 , λ1 , . . . , λ2n = λ), we will recursively define a sequence
(P0 , T0 ), (P1 , T1 ),. . . , (P2n , T2n ) where Pi is a set of ordered pairs of integers in [n],
and Ti is an SYT of shape λi . Let P0 be the empty set, and let T0 be the empty
SYT (on the empty alphabet).
(1) If λi = λi−1 , then (Pi , Ti ) = (Pi−1 , Ti−1 ).
(2) If λi ⊃ λi−1 , then i = 2k for some integer k ∈ [n]. In this case let Pi = Pi−1
and Ti is obtained from Ti−1 by adding the entry k in the square λi \ λi−1 .
(3) If λi ⊂ λi−1 , then i = 2k − 1 for some integer k ∈ [n]. In this case let Ti
be the unique SYT (on a suitable alphabet) of shape λi such that Ti−1 is
obtained from Ti by row-inserting some number j. Note that j must be less
than k. Let Pi be obtained from Pi−1 by adding the ordered pair (j, k).
It is clear from the above construction that (i) P0 ⊆ P1 ⊆ · · · ⊆ P2n , (ii) for each
integer i, it appears at most once as the first component of an ordered pair in P2n ,
and appears at most once as the second component of an ordered pair in P2n . Let
ψ(V ) = (P, T2n ), where P is the partition on [n] whose standard representation is
P2n .
Note that if an integer i appears in T2n , then P2n can not contain any ordered
pair (i, j) with i < j. It follows that i is the maximal element in the block containing
it. Hence the content of T2n is a subset of max(P ).
Example 2.5. As an example of the map ψ, let the vacillating tableau be
∅, ∅, 1, 1, 2, 2, 2, 2, 21, 21, 211, 21, 21, 11, 21.
6
W. CHEN, E. DENG, R. DU, R. STANLEY, AND C. YAN
Then the pairs (Bi , Ti ) (where Bi is the pair added to Pi−1 to obtain Pi ) are given
by
0
∅
i
Ti
Bi
1
∅
2 3 4
5
6 7 8 9 10 11
12 13
1 1 12 12 12 12 12 12 12 14
14 1
4 4 4 5
5 5
5
(2, 6)
(4, 7)
14
17
5
Hence
7
T = 1
5 ,
P = 1-26-3-47-5.
The map ψ is bijective since the above construction can be reversed. Given a
pair (P, T ), where P is a partition of [n] and T is an SYT whose content consists
of maximal elements of some blocks of P , let E(P ) be the standard representation
of P , and T2n = T . We work our way backwards from T2n , reconstructing the
preceding tableaux and hence the sequence of shapes. If we have the SYT T2k for
some k ≤ n, we can get the tableaux T2k−1 , T2k−2 by the following rules.
(1) T2k−1 = T2k if the integer k does not appear in T2k . Otherwise T2k−1 is
obtained from T2k by deleting the square containing k.
(2) T2k−2 = T2k−1 if E(P ) does not have an edge of the form (i, k). Otherwise
there is a unique i < k such that (i, k) ∈ E(P ). In that case let T2k−2 be
obtained from T2k−1 by row-inserting i, or equivalently, T2k−2 = (T2k−1 ←−
i).
Proof of Theorem 2.4. Part (i) follows from the bijection ψ, where a block
of P is distinguished if its maximal element belongs to content(T ). For part (ii),
simply note that to get a structure counted by B(n, k), we can partition [n] into
two subsets, S and T , and then partition S into k blocks and put a mark on each
block, and partition T arbitrarily. The generating function of B(n, k) then follows
from the well-known generating functions for S(n, k), the Stirling number of the
second kind, and for the Bell number B(n),
X
X
xn
1
xn
S(n, k)
= (ex − 1)k ,
B(n)
= exp(ex − 1).
n!
k!
n!
n≥0
n≥k
Remark 2.6. (1). Restricting to vacillating tableaux of empty shape, the map ψ
provides a bijection between the set V∅2n of vacillating tableaux of empty shape and
length 2n and the set of partitions of [n]. In particular, g∅ (n), the cardinality of
V∅2n , is equal to the nth Bell number B(n).
(2). Note that there is a symmetry between the four types of movements in the
definition of vacillating tableaux. Thus any walk from ∅ to ∅ in m + n steps can be
viewed as a walk from ∅ to some shape λ in n steps, then followed by the reverse
of a walk from ∅ to λ in m steps. It follows that
X
(2.3)
gλ (n)gλ (m) = g∅ (m + n) = B(m + n).
λ
For the case m = n = k, the identity (2.3) is proved by Halverson and Lowandowski
[16], who gave a bijective proof using similar procedures as those in ψ.
(3). The partition algebra Pn is a certain semisimple algebra, say over C, whose
dimension is the Bell number B(n) (the number of partitions of [n]). (The algebra
Pn depends on a parameter x which is irrelevant here.) See [15, 16] for a survey
CROSSINGS AND NESTINGS
7
of this topic. Vacillating tableaux are related to irreducible representations of Pn
in the same way that SYT of content [n] are related to irreducible representations
of the symmetric group Sn . In particular, the irreducible representations Iλ of Pn
are indexed by partitions λ for which there exists a vacillating tableau of shape λ
and length 2n, and dim In is the number of such vacillating tableaux. This result is
equivalent to [15, Thm. 2.24(b)], but that paper does not explicitly define the notion
of vacillating tableau. Combinatorial identities arising from partition algebra and
its subalgebras are discussed in [16], where the authors used the notion of vacillating
tableau after the distribution of a preliminary version of this paper.
3. Crossings and Nestings of Partitions
In this section we restrict the map ψ to vacillating tableaux of empty shape,
for which ψ provides a bijection between the set of vacillating tableaux of empty
shape and length 2n and the set of partitions of [n]. To make the bijection clear,
we restate the inverse map from the set of partitions to vacillating tableaux.
The Map φ from partitions to vacillating tableaux. Given a partition P ∈ Πn
with the standard representation, we construct the sequence of SYT’s, hence the
vacillating tableau φ(P ) as follows: Start from the empty SYT by letting T2n = ∅,
read the number j ∈ [n] one by one from n to 1, and define T2j−1 , T2j−2 for each
j. There are four cases.
1. If j is the righthand endpoint of an arc (i, j), but not a lefthand endpoint, first
do nothing, then insert i (by the RSK algorithm) into the tableau.
2. If j is the lefthand endpoint of an arc (j, k), but not a righthand endpoint, first
remove j, then do nothing.
3. If j is an isolated point, do nothing twice.
4. If j is the righthand endpoint of an arc (i, j), and the lefthand endpoint of
another arc (j, k), then delete j first, and then insert i.
The vacillating tableau φ(P ) is the sequences of shapes of the above SYT’s.
Example 3.1. Let P be the partition 1457-26-3 of [7].
t
t
t
t
t
t
t
1
2
3
4
5
6
7
Figure 1. The standard representation of the partition 1457-26-3.
Starting from ∅ on the right, go from 7 to 1, the seven steps are (1) do nothing,
then insert 5, (2) do nothing, then insert 2, (3) delete 5 and insert 4, (4) delete 4
and insert 1, (5) do nothing twice, (6) remove 2 then do nothing, (7) remove 1 then
do nothing. Hence the corresponding SYT’s, constructed from right to left, are
∅
∅
1
1
1 1 1 2
2 2 2
24
2
2 5
5
5
∅
∅
The vacillating tableau is
∅, ∅, 1, 1, 11, 11, 11, 1, 2, 1, 11, 1, 1, ∅, ∅.
The relation between cr(P ), ne(P ) and the vacillating tableau is given in the
next theorem.
8
W. CHEN, E. DENG, R. DU, R. STANLEY, AND C. YAN
Theorem 3.2. Let P ∈ Πn and φ(P ) = (∅ = λ0 , λ1 , . . . , λ2n = ∅). Then cr(P ) is
the most number of rows in any λi , and ne(P ) is the most number of columns in
any λi .
Proof. We prove Theorem 3.2 in four steps. First, we interpret a k-crossing/knesting of P in terms of entries of SYT’s Ti in φ(P ). Then, we associate to each
SYT Ti a sequence σi whose terms are entries of Ti . We prove that Ti is the
insertion tableau of σi under the RSK algorithm, and apply Schensted’s theorem
to conclude the proof.
Step 1. Let T (P ) = (T0 , T1 , . . . , T2n ) be the sequence of SYT’s associated to
the vacillating tableau φ(P ). By the construction of ψ and φ, a pair (i, j) is an
arc in the standard representation of P if and only if i is an entry in the SYT’s
T2i , T2i+1 , . . . , T2j−2 . We say that the the integer i is added to T (P ) at step i and
leaves at step j.
First we prove that the arcs (i1 , j1 ), . . . , (ik , jk ) form a k-crossing of P if and only
if there exists a tableau Ti in T (P ) such that the integers i1 , i2 , . . . , ik ∈ content(Ti ),
and i1 , i2 , . . . , ik leave T (P ) in increasing order according to their numerical values.
Given a k-crossing ((i1 , j1 ), . . . , (ik , jk )) of P , where ir < jr for 1 ≤ r ≤ k and
i1 < i2 < · · · < ik < j1 < j2 < · · · < jk , the integer ir is added to T (P ) at step ir
and leaves at step jr . Hence all ir are in T2j1 −2 , and they leave T (P ) in increasing
order. The converse is also true: if there are k integers i1 < i2 < · · · < ik all
appearing in the same tableau at some steps, and then leave in increasing order,
say at steps j1 < j2 < · · · < jk , then ik < j1 and the pairs ((i1 , j1 ), . . . , (ik , jk )) ∈
P form a k-crossing. By a similar argument arcs (i1 , j1 ), . . . , (ik , jk ) form a knesting of P if and only if there exists a tableaux Ti in T (P ) such that the integers
i1 , i2 , . . . , ik ∈ content(Ti ), and i1 , i2 , . . . , ik leave T (P ) in decreasing order.
Step 2. For each Ti ∈ T (P ), we define a permutation σi of content(Ti ) (backward) recursively as follows. Let σ2n be the empty sequence. (1) If Ti = Ti−1 , then
σi−1 = σi . (2) If Ti−1 is obtained from Ti by row-inserting some number j, then
σi−1 = σi j, the juxtaposition of σi and j. (3) If Ti is obtained from Ti−1 by adding
the entry i/2, (where i must be even), then σi−1 is obtained from σi by deleting
the number i/2. Note that in the last case, i/2 must be the largest entry in σi .
Clearly σi is a permutation of the entries in content(Ti ). If σi = w1 w2 · · · wr ,
then the entries of content(Ti ) leave T (P ) in the order wr , . . . , w2 , w1 .
RSK
Step 3. Claim: If σi 7−→ (Ai , Bi ), then Ai = Ti .
We prove the claim by backward induction. The case i = 2n is trivial as both
A2n and T2n are the empty SYT. Assume the claim is true for some i, 1 ≤ i ≤ 2n.
We prove that the claim holds for i − 1.
If Ti−1 = Ti , the claim holds by the inductive hypothesis. If Ti−1 is obtained
from Ti by inserting the number j, then the claim holds by the definition of the
RSK algorithm. It is only left to consider the case that Ti−1 is obtained from Ti
by removing the entry j = i/2.
Let us write σi as u1 u2 · · · us jv1 · · · vt , and σi−1 as u1 u2 · · · us v1 · · · vt , where
j > u1 , . . . , us , v1 , . . . , vt . We need to show that the insertion tableau of σi−1 is
the same as the insertion tableau of σi deleting the entry j, i.e., Ai−1 = Ai \{j}.
Proof by induction on t. If t = 0 then it is true by the RSK algorithm that Ai
is obtained from Ai−1 by adding j at the end of the first row. Assume it is true
for t − 1, i.e., A(u1 · · · us v1 · · · vt−1 ) = A(u1 · · · us jv1 · · · vt−1 ) \ {j}. Note that
in A(u1 · · · us jv1 · · · vt−1 ), if j is in position (x, y), then there is no element in
CROSSINGS AND NESTINGS
9
positions (x, y + 1) or (x + 1, y). Now we insert the entry vt by the RSK algorithm.
Consider the insertion path I = I(A(u1 · · · us jv1 · · · vt−1 ) ←− vt ). If j does not
appear on this path, then we would have the exact same insertion path when
inserting vt into A(u1 · · · us v1 · · · vt−1 ). This insertion path results in the same
change to A(u1 · · · us jv1 · · · vt−1 ) and A(u1 · · · us v1 · · · vt−1 ), which does not touch
the position (x, y) of j. So A(u1 · · · us v1 · · · vt−1 vt ) = A(u1 · · · us jv1 · · · vt−1 vt )\{j}.
On the other hand, if j appears in the insertion path I, i.e., (x, y) ∈ I, then since j
is the largest element, it must be bumped into the (x + 1)-th row, and become the
last entry in the (x + 1)-th row without bumping any number further. Then the
insertion path of vt into A(u1 · · · us v1 · · · vt−1 ) is I minus the last position {(x +
1, ∗)}, and again we have A(u1 · · · us v1 · · · vt−1 vt ) = A(u1 · · · us jv1 · · · vt−1 vt ) \ {j}.
This finishes the proof of the claim.
Step 4. We shall need the following theorem of Schensted [22][25, Thms. 7.23.13,
7.23.17], which gives the basic connection between the RSK algorithm and the
increasing and decreasing subsequences.
Schensted’s Theorem Let σ be a sequence of integers whose
RSK
terms are distinct. Assume σ 7−→ (A, B), where A and B are
SYT’s of the shape λ. Then the length of the longest increasing
subsequence of σ is λ1 (the number of columns of λ), and the length
of the longest decreasing subsequence is λ01 (the number of rows of
λ).
Now we are ready to prove Theorem 3.2. By Steps 1 and 2, a partition P has a
k-crossing if and only if there exists i such that σi has a decreasing subsequence of
length k. The claim in Step 3 implies that the shape of the sequence σi is exactly
the diagram of the i-th partition λi in the vacillating tableau φ(P ). By Schensted’s
Theorem, σi has a decreasing subsequence of length k if and only if the partition λi
in φ(P ) has at least k rows. This proves the statement for cr(P ) in Theorem 3.2.
The statement for ne(P ) is proved similarly. The symmetric joint distribution of statistics cr(P ) and ne(P ) over Pn (S, T )
follows immediately from Theorem 3.2.
Proof of Theorem 1.
From Theorem 3.2, a partition P ∈ Πn has cr(P ) = k and ne(P ) = j if and only
if for the partitions {λi }2n
i=0 of the vacillating tableau φ(P ), the maximal number
of rows of the diagram of any λi is k, and the maximal number of columns of the
diagram of any λi is j. Let τ be the involution defined on the set V∅2n by taking the
conjugate to each partition λi . For i ∈ [n], i ∈ min(P ) (resp. max(P )) if and only
if λ2i−1 = λ2i−2 and λ2i \ λ2i−1 = , (reps. λ2i−2 \ λ2i−1 = and λ2i = λ2i−1 ).
Since τ preserves min(P ) and max(P ), it induces an involution on Pn (S, T ) which
exchanges the statistics cr(P ) and ne(P ). This proves Theorem 1.1. Let λ = (λ1 , λ2 , . . . ) be the shape of a sequence w of distinct integers. Schensted’s
Theorem provides a combinatorial interpretation of the terms λ1 and λ01 : they are
the length of the longest increasing and decreasing subsequences of w. In [14] C.
Greene extended Schensted’s Theorem by giving an interpretation of the rest of the
diagram of λ = (λ1 , λ2 , . . . ).
Assume w is a sequence of length n. For each k ≤ n, let dk (w) denote the length
of the longest subsequence of w which has no increasing subsequences of length
k + 1. It can be shown easily that any such sequence is obtained by taking the
10
W. CHEN, E. DENG, R. DU, R. STANLEY, AND C. YAN
union of k decreasing subsequences. Similarly, define ak (w) to be the length of the
longest subsequence consisting of k ascending subsequences.
Theorem 3.3 (Greene). For each k ≤ n,
ak (w)
= λ1 + λ2 + · · · + λk ,
dk (w)
= λ01 + λ02 + · · · + λ0k ,
where λ0 = (λ01 , λ02 , . . . ) is the conjugate of λ. We may consider the analogue of Greene’s Theorem for set partitions. Let P ∈
Πn with the standard representation {(i1 , j1 ), (i2 , j2 ), . . . , (it , jt )}, where ir < jr for
1 ≤ r ≤ t. Let er = (ir , jr ). We define the crossing graph Cr(P ) of P as follows.
The vertex set of Cr(P ) is {e1 , e2 , . . . , et }. Two arcs er and es are adjacent if and
only if the edges er and es are crossing, that is, ir < is < jr < js . Clearly a
k-crossing of P corresponds to a k-clique of Cr(P ). Let crr (P ) be the maximal
number of vertices in a union of r cliques of Cr(P ). In other words, crr (P ) is the
maximal number of arcs in a union of r crossings of P . Similarly, let Ne(P ) be the
graph defined on the vertex set {e1 , . . . , et } where two arcs er and es are adjacent
if and only if ir < is < js < jr . Let ner (P ) be the maximal number of vertices in a
union of r cliques of Ne(P ). In other words, ner (P ) is the maximal number of arcs
in a union of r nestings of P .
Proposition 3.4. Let P = ((i1 , j1 ), (i2 , j2 ), . . . , (it , jt )) be the standard representation of a partition of [n], where ir < jr for all 1 ≤ r ≤ t and j1 < j2 < · · · < jt . Let
α(P ) be the sequence i1 i2 · · · it . Then there is a one-to-one correspondence between
the set of nestings of P and the set of decreasing subsequences of α(P ).
Proof. Let φ(P ) be the vacillating tableau corresponding to P , and T (P ) the
sequence of SYT’s constructed in the bijection. Then α(P ) records the order in
which the entries of Ti ’s leave T (P ). Let σ = it · · · i2 i1 be the reverse of α(P ), and
{σi : 1 ≤ i ≤ 2n} the permutation of content(Ti ) defined in Step 2 of the proof of
Theorem 3.2. Then the σi ’s are subsequences of σ.
From Steps 1 and 2 of the proof of Theorem 3.2, nestings of P are represented
by the increasing subsequences of σi , 1 ≤ i ≤ 2n, and hence by the increasing
subsequences of σ. Conversely, let ir1 < ir2 < · · · < irt be an increasing subsequence
of σ. Being a subsequence of σ means that its terms leave T (P ) in reverse order,
so irt leaves first in step jrt . Thus all the entries ir1 , . . . , irt appear in the SYT’s
Ti with 2irt ≤ i ≤ 2jrt − 2. Therefore ir1 ir2 · · · irt is also an increasing subsequence
of σi , for 2irt ≤ i ≤ 2jrt − 2. Combining Proposition 3.4 and Greene’s Theorem, we have the following corollary describing net (P ).
Corollary 3.5. Let P and α(P ) be as in Proposition 3.4. Then
ner (P ) = λ01 + λ02 + · · · + λ0r ,
where λ is the shape of α(P ), and λ0 is the conjugate of λ. The situation for crr (P ) is more complicated. We don’t have a result similar to Proposition 3.4. Any crossing of P uniquely corresponds to an increasing
subsequence of α(P ). But the converse is not true. An increasing subsequence
ir1 ir2 · · · irt corresponds to a t-crossing of P only if we have the additional condition irt < jr1 . It would be interesting to get a result for crr (P ) analogous to
Corollary 3.5.
CROSSINGS AND NESTINGS
11
To conclude this section we discuss the enumeration of noncrossing partitions.
The following theorem is a direct corollary of the bijection between vacillating
tableaux and partitions.
Theorem 3.6. Let i denote the ith unit coordinate vector in Rk−1 . The number
of k-noncrossing partitions of [n] equals the number of closed lattice walks in the
region
Vk = {(a1 , a2 , . . . , ak−1 ) : a1 ≥ a2 ≥ · · · ≥ ak−1 ≥ 0, ai ∈ Z}
from the origin to itself of length 2n with steps ±i or (0, 0, . . . , 0), with the property
that the walk goes backwards (i.e., with step −i ) or stands still (i.e., with step
(0, 0, . . . , 0)) after an even number of steps, and goes forwards (i.e., with step +i )
or stands still after an odd number of steps. Recall that a partition P ∈ Πn is k-noncrossing if cr(P ) < k, and is k-nonnesting
if ne(P ) < k. A partition P has no k-crossings and no j-nestings if and only if for
all the partitions λi of φ(P ), the diagram fits into a (k − 1) × (j − 1) rectangle.
Taking the conjugate of each partition, we get bijective proofs of Corollaries 1.2
and 1.3.
Theorem 1.1 asserts the symmetric distribution of cr(P ) and ne(P ) over Pn (S, T ),
for all S, T ⊆ [n] with |S| = |T |. Not every Pn (S, T ) is nonempty. A set Pn (S, T ) is
nonempty if and only if for all i ∈ [n], |S ∩ [i]| ≥ |T ∩ [i]|. Another way to describe
the nonempty Pn (S, T ) is to use lattice paths. Associate to each pair (S, T ) a lattice
path L(S, T ) with steps (1, 1), (1, −1) and (1, 0): start from (0, 0), read the integers
i from 1 to n one by one, and move two steps for each i.
1. If i ∈ S ∩ T , move (1, 0) twice.
2. If i ∈ S \ T , move (1, 0) then (1, 1).
3. If i ∈ T \ S, move (1, −1) then (1, 0).
4. If i ∈
/ S ∪ T , move (1, −1) then (1, 1).
This defines a lattice path L(S, T ) from (0, 0) to (2n, 0), Conversely, the path
uniquely determines (S, T ). Then Pn (S, T ) is nonempty if and only if the lattice
path L(S, T ) is a Motzkin path, i.e., never goes below the x-axis.
There are existing notions of noncrossing partitions and nonnesting partitions,
e.g., [25, Ex.6.19]. A noncrossing partition of [n] is a partition of [n] in which no
two blocks “cross” each other, i.e., if a < b < c < d and a, c belong to a block B and
b, d to another block B 0 , then B = B 0 . A nonnesting partition of [n] is a partition
of [n] such that if a, e appear in a block B and b, d appear in a different block B 0
where a < b < d < e, then there is a c ∈ B satisfying b < c < d.
It is easy to see that P is a noncrossing partition if and only if the standard
representation of P has no 2-crossing, and P is a nonnesting partition if and only
if the standard representation of P has no 2-nesting. Hence the vacillating tableau
correspondence, in the case of 1-row/column tableaux, gives bijections between
noncrossing partitions of [n], nonnesting partitions of [n], (both are counted by
Catalan numbers) and sequences 0 = a0 , a1 , ..., a2n = 0 of nonnegative integers
such that a2i+1 = a2i or a2i − 1, and a2i = a2i−1 or a2i−1 + 1. These sequences
a0 , ..., a2n give a new combinatorial interpretation of Catalan numbers.
Replacing a term ai+1 = ai + 1 with a step (1, 1), a term ai+1 = ai − 1 with
a step (1, −1), and a term ai+1 = ai with a step (1, 0), we get a Motzkin path,
so we also have a bijection between noncrossing/nonnesting partitions and certain
Motzkin paths. The Motzkin paths are exactly the ones defined as L(S, T ), where
12
W. CHEN, E. DENG, R. DU, R. STANLEY, AND C. YAN
S = min(P ) and T = max(P ). Conversely, given a Motzkin path of the form
L(S, T ), we can recover uniquely a noncrossing partition and a nonnesting partition.
Write the path as {(i, ai ) : 0 ≤ i ≤ 2n}. Let A = [n] \ T and B = [n] \ S.
Clearly |A| = |B|. Assume A = {i1 , i2 , . . . , it }< , and B = {j1 , j2 , . . . , jt }< where
elements are listed in increasing order. Then to get the standard representation of
the noncrossing partition, pair each jr with max{is ∈ A : is < jr , a2is = a2jr −2 }.
To get the standard representation of the nonnesting partition, pair each jr with
ir , for 1 ≤ r ≤ t.
Remark 3.7. In our definition, a k-crossing is defined as a set of k mutually crossing arcs in the standard representation of the partition. There exist some other
definitions. For example, in [17] M. Klazar defined 3-noncrossing partition as a
partition P which does not have 3 mutually crossing blocks. It can be seen that
P is 3-noncrossing in Klazar’s sense if and only if there do not exist 6 elements
a1 < b1 < c1 < a2 < b2 < c2 in [n] such that a1 , a2 ∈ A, b1 , b2 ∈ B, c1 , c2 ∈ C, and
A, B, C are three distinct blocks of P .
Klazar’s definition of 3-noncrossing partitions is different from ours. For example,
let P be the partition 15-246-37 of [7], with standard representation as follows:
r r r r r r r
1 2 3 4 5 6 7
According to Klazar’s definition, P has a 3-crossing, since we have 1 < 2 < 3 <
5 < 6 < 7 and {1, 5}, {2, 6} and {3, 7} belong to three different blocks, respectively.
On the other hand, P has no 3-crossing on our sense.
For general k, these three notions of k-noncrossing partitions, i.e., (1) no kcrossing in the standard representation of P , (2) no k mutually crossing arcs in
distinct blocks of P , and (3) no k mutually crossing blocks, are all different, with
the first being the weakest, and the third the strongest.
4. A Variant: Partitions and Hesitating Tableaux
We may also consider the enhanced crossing/nesting of a partition, by taking
isolated points into consideration. For a partition P of [n], let the enhanced representation of P be the union of the standard representation of P and the loops
{(i, i) : i is an isolated point of P }. An enhanced k-crossing of P is a set of k
edges (i1 , j1 ), (i2 , j2 ), . . . , (ik , jk ) of the enhanced representation of P such that
i1 < i2 < · · · < ik ≤ j1 < j2 < · · · < jk . In particular, two arcs of the form (i, j)
and (j, l) with i < j < l are viewed as crossing. Similarly, an enhanced k-nesting
of P is a set of k edges (i1 , j1 ), (i2 , j2 ), . . . , (ik , jk ) of the enhanced representation
of P such that i1 < i2 < · · · < ik ≤ jk < · · · < j2 < j1 . In particular, an edge (i, k)
and an isolated point j with i < j < k form an enhanced 2-nesting.
Let cr(P ) be the size of the largest enhanced crossing, and ne(P ) the size of
the largest enhanced nesting. Using a variant form of vacillating tableau, we again
obtain a symmetric joint distribution of the statistics cr(P ) and ne(P ).
The variant tableau is a hesitating tableau of shape ∅ and length 2n, which is a
path on the Hasse diagram of Young’s lattice from ∅ to ∅ where each step consists of
a pair of moves, where the pair is either (i) doing nothing then adding a square, (ii)
removing a square then doing nothing, or (iii) adding a square and then removing
a square.
CROSSINGS AND NESTINGS
13
Example 4.1. There are 5 hesitating tableaux of shape ∅ and length 6. They are
(4.1)
∅ 1
∅ 1
∅ ∅
∅ ∅
∅ ∅
∅ 1
∅ ∅
1 11
1 ∅
1 2
∅
1
1
∅
1
1 ∅
∅ ∅
∅ ∅
1 ∅
∅ ∅
To see the equivalence with vacillating tableaux, let U be the operator that takes
a shape to the sum of all shapes that cover it in Young’s lattice (i.e., by adding a
square), and similarly D takes a shape to the sum of all shapes that it covers in
Young’s lattice (i.e., by deleting a square). Then, as is well-known, DU − U D = I
(the identity operator). See, e.g. [23][25, Exer. 7.24]. It follows that
(4.2)
(U + I)(D + I) = DU + ID + U I.
Iterating the left-hand side generates vacillating tableaux, and iterating the righthand side gives the hesitating tableaux defined above.
A bijective map between partitions of [n] and hesitating tableaux of empty shape
has been given by Korn [18], based on growth diagrams. Here by modifying the
map φ defined in Section 3 we get a more direct bijection φ̄ between partitions and
hesitating tableaux of empty shape, which leads to the symmetric joint distribution
of cr(P ) and ne(P ). The construction and proofs are very similar to the ones given
in Sections 2 and 3, and hence are omitted here. We will only state the definition
of the map φ̄ from partitions to hesitating tableaux, to be compared with the map
φ in Section 3.
The Bijection φ̄ from Partitions to Hesitating Tableaux.
Given a partition P ∈ Πn with the enhanced representation, we construct the
sequence of SYT’s, and hence the hesitating tableau φ̄(P ), as follows: Start from
the empty SYT by letting T2n = ∅, read the numbers j ∈ [n] one by one from n
to 1, and define two SYT’s T2j−1 , T2j−2 for each j. When j is a lefthand endpoint
only, or a righthand endpoint only, the construction is identical to that of the map
φ. Otherwise,
1. If j is an isolated point, first insert j, then delete j.
2. If j is the righthand endpoint of an arc (i, j), and the lefthand endpoint of
another arc (j, k), then insert i first, and then delete j.
Example 4.2. For the partition 1457-26-3 of [7] in Figure 2, the corresponding
SYT’s are
∅ ∅ 1 1 1 13 1 14 24 24 2 5 5 ∅ ∅
2 2 2 2
5 5
The hesitating tableau φ̄(P ) is
∅, ∅, 1, 1, 11, 21, 11, 21, 2, 21, 11, 1, 1, ∅, ∅.
j
t
t
t
t
t
t
t
1
2
3
4
5
6
7
Figure 2. The enhanced representation of the partition 1457-26-3.
14
W. CHEN, E. DENG, R. DU, R. STANLEY, AND C. YAN
The conjugation of shapes does not preserve min(P ) or max(P ). Instead, it
preserves min(P )\max(P ), and max(P )\min(P ). Let S, T be disjoint subsets of [n]
with the same cardinality, P̄n (S, T ) = {P ∈ Πn : min(P )\max(P ) = S, max(P )\
min(P ) = T }, and f¯n,S,T (i, j) = #{P ∈ P̄n (S, T ) : cr(P ) = i, ne(P ) = j}.
Theorem 4.3. We have
f¯n,S,T (i, j) = f¯n,S,T (j, i).
As a consequence, Corollaries 1.2 and 1.3 remain valid if we define k-noncrossing
(or k-nonnesting) by cr(P
¯ ) < k (or ne(P
¯ ) < k).
Remark 4.4. As done for vacillating tableaux, one can extend the definition of
hesitating tableaux by considering moves from ∅ to λ, and denote by fλ (n) the
number of such hesitating tableaux of length 2n. Identity (4.2) implies that fλ (n) =
gλ (n), the number of vacillating tableaux from ∅ to λ. It follows that f∅ (n) = B(n),
and
X
fλ (n)fλ (m) = B(m + n),
λ
where B(n) is the nth Bell number. For further discussion of the number fλ (n),
see [26, Problem 33] (version of 17 August 2004).
5. Enumeration of k-Noncrossing Matchings
Restricting Theorem 1.1 to disjoint subsets (S, T ) of [n], where n = 2m and
|S| = |T | = m, we get the symmetric joint distribution of the crossing number and
nesting number for matchings, as stated in Corollary 1.4 in Section 1.
In a complete matching, an integer is either a left endpoint or a right endpoint
in the standard representation. In applying the map φ to complete matchings
on [2m], if we remove all steps which do nothing, we obtain a sequence ∅ = λ0 ,
λ1 , . . . , λ2m = ∅ of partitions such that for all 1 ≤ i ≤ 2m, the diagram of λi is
obtained from that of λi−1 by either adding one square or removing one square.
Such a sequence is called an oscillating tableau (or up-down tableau) of empty
shape and length 2m. Thus we get a bijection between complete matchings on
[2m] and oscillating tableaux of empty shape and length 2m. This bijection was
originally constructed by the fourth author, and then extended by Sundaram [27]
to arbitrary shapes to give a combinatorial proof of the Cauchy identity for the
symplectic group Sp(2m). The explicit description of the bijection has appeared
in [27] and was included in [25, Exercise 7.24]. Oscillating tableaux first appeared
(though not with that name) in [5].
Recall that the ordinary RSK algorithm gives a bijection between the symmetric
group Sm and pairs (P, Q) of SYTs of the same shape λ ` m. This result and
the Schensted’s theorem can be viewed as a special case of what we do. Explicitly,
identify an SYT T of shape λ and content [m] with a sequence ∅ = λ0 , λ1 , . . . , λm =
λ of integer partitions where λi is the shape of the SYT T i obtained from T by
deleting all entries {j : j > i}. Let w be a permutation of [m], and form the
matching Mw on [2m] with arcs between w(i) and 2m − i + 1. We get an oscillating
tableau Ow that increases to the shape λ ` m and then decreases to the empty
RSK
shape. Assume w 7−→ (A(w), B(w)), where A(w) is the (row)-insertion tableau
and B(w) the recording tableau. Then A(w) is given by the first m steps of Ow ,
and B(w) the reverse of the last m steps. The size of the largest crossing (resp.
CROSSINGS AND NESTINGS
15
nesting) of Mw is exactly the length of the longest decreasing (resp. increasing)
subsequence of w.
Example 5.1. Let w = 231. Then
A(w) =
1
2
3
,
B(w) =
1
3
2
.
The matching Mw and the corresponding oscillating tableau are given below.
r
1
∅
r
2
1
r
3
11
r
4
21
r
5
2
r
6
1
∅
Remark 5.2. The Brauer algebra Bm is a certain semisimple algebra, say over C,
whose dimension is the number 1·3·· · · (2m−1) of matchings on [2m]. (The algebra
Bm depends on a parameter x which is irrelevant here.) Oscillating tableaux of
length 2m are related to irreducible representations of Bm in the same way that
SYT of content [m] are related to irreducible representations of the symmetric group
Sm and that vacillating tableaux of length 2m are related to irreducible representations of the partition algebra Pm . In particular, the irreducible representations
Jλ of Bm are indexed by partitions λ for which there exists an oscillating tableau
of shape λ and length 2m, and dim Jλ is the number of such oscillating tableaux.
See e.g. [4, Appendix B6] for further information.
Next we use the bijection between complete matchings and oscillating tableaux to
study the enumeration of k-noncrossing matchings. All the results in the following
hold for k-nonnesting matchings as well.
For complete matchings, Theorem 3.6 becomes the following.
Corollary 5.3. The number of k-noncrossing matchings of [2m] is equal to the
number of closed lattice walks of length 2m in the set
Vk = {(a1 , a2 , . . . , ak−1 ) : a1 ≥ a2 ≥ · · · ≥ ak−1 ≥ 0, ai ∈ Z}
from the origin to itself with unit steps in any coordinate direction or its negative. Restricted to the cases k = 2, 3, Corollary 5.3 leads to some nice combinatorial
correspondences. Recall that a Dyck path of length 2m is a lattice path in the
plane from the origin (0, 0) to (2m, 0) with steps (1, 1) and (1, −1), that never
passes below the x-axis. A pair (P, Q) of Dyck paths is noncrossing if they have
the same origin and the same destination, and P never goes below Q.
Corollary 5.4.
(1) The set of 2-noncrossing matchings is in one-to-one correspondence with the set of Dyck paths.
(2) The set of 3-noncrossing matchings is in one-to-one correspondence with
the set of pairs of noncrossing Dyck paths.
Proof. By Corollary 5.3, 2-noncrossing matchings are in one-to-one correspondence
with closed lattice paths {~vi = (xi )}2m
i=0 with x0 = x2m = 0, xi ≥ 0 and xi+1 − xi =
±1. Given such a 1-dimensional lattice path, define a lattice path in the plane by
letting P = {(i, xi ) | i = 0, 1, . . . , 2m}. Then P is a Dyck path, and this gives the
desired correspondence.
For k = 3, 3-noncrossing matchings are in one-to-one correspondence with 2dimensional lattice paths {~vi = (xi , yi )}2m
i=0 with (x0 , y0 ) = (x2m , y2m ) = (0, 0),
16
W. CHEN, E. DENG, R. DU, R. STANLEY, AND C. YAN
xi ≥ yi ≥ 0 and (xi+1 , yi+1 )−(xi , yi ) = (±1, 0) or (0, ±1). Given such a lattice path,
define two lattice paths in the plane by setting P = {(i, xi + yi ) | i = 0, 1, . . . , 2m}
and Q = {(i, xi − yi ) | i = 0, 1, . . . , 2m}. Then (P, Q) is a pair of noncrossing Dyck
paths. It is easy to see that this is a bijection. Example 5.5. We illustrate the bijections between 3-noncrossing matchings, oscillating tableaux, and pairs of noncrossing Dyck paths. The oscillating tableau
is
∅, 1, 2, 21, 31, 21, 11, 21, 2, 1, ∅.
The sequence of SYT’s as defined in the bijection is
2 1 2 4 , 1 2 , 1 , 1 7 , 37, 3, ∅.
∅, 1, 12, 1
3 , 3
3
3 3
The corresponding matching and the pair of noncrossing Dyck paths are given in
Figure 3.
r
r
r
r
r
@
@r
r
@
@r
@
@r
r
@
@r
@
@r
r
@
@r
r@
@r
@r@
r @
@r
@
@r
r
r r r r r r r r r r
r
1 2 3 4 5 6 7 8 9 10
Figure 3. The matching and the pair of noncrossing Dyck paths.
Let fk (m) be the number of k-noncrossing matchings of [2m]. By Corollary 5.3
it is also the number of lattice paths of length 2m in the region Vk from the origin
to itself with step set {±1 , ±2 , . . . , ±k−1 }. Set
Fk (x) =
X
fk (m)
m
x2m
.
(2m)!
It turns out that a determinantal expression for Fk (x) has been given by Grabiner
and Magyar [12]. It is simply the case λ = η = (m, m − 1, ..., 1) of equation (38) in
[12], giving
(5.1)
k−1
Fk (x) = det [Ii−j (2x) − Ii+j (2x)]i,j=1 ,
where
Im (2x) =
X
j≥0
xm+2j
,
j!(m + j)!
the hyperbolic Bessel function of the first kind of order m [30]. One can easily check
that when k = 2, the generating function of 2-noncrossing matchings equals
F2 (x) = I0 (2x) − I2 (2x) =
X
j≥0
Cj
x2j
,
(2j)!
CROSSINGS AND NESTINGS
17
where Cj is the j-th Catalan number. When k = 3, we have
f3 (m)
3!(2m + 2)!
m!(m + 1)!(m + 2)!(m + 3)!
=
2
= Cm Cm+2 − Cm+1
.
This result agrees with the formula on the number of pairs of noncrossing Dyck
paths due to Gouyou-Beauchamps in [11].
Remark 5.6. The determinant formula (5.1) has been studied by Baik and Rains in
[2, Eqs. (2.25)]. One simply puts i − 1 for j and j − 1 for k in (2.25) of [2] to get our
formula. The same formula was also obtained by Goulden [10] as the generating
function for fixed point free permutations with no decreasing subsequence of length
greater than 2k. See Theorem 1.1 and 2.3 of [10] and specialize hi to be xi /i!, so
gl becomes the hyperbolic Bessel function. The asymptotic distribution of cr(M )
follows from another result of Baik and Rains. In Theorem 3.1 of [3] they obtained
the limit distribution for the length of the longest decreasing subsequence of fixed
point free involutions w. But representing w as a matching M , the condition that
w has no decreasing subsequence of length 2k + 1 is equivalent to the condition
that M has no k + 1-nesting, and we already know that cr(M ) and ne(M ) have the
same distribution. Combining the above results, one has
!
√
cr(M ) − 2m
x
lim Pr
= F1 (x),
≤
m→∞
2
(2m)1/6
where
Z ∞
1
u(s)ds ,
2 x
where F (x) is the Tracy-Widom distribution and u(x) the Painlevé II function.
F1 (x) =
p
F (x) exp
Similarly one can try to enumerate complete matchings of [2m] with no (k + 1)crossing and no (j + 1)-nesting. By the oscillating tableau bijection this is just
the number of walks of length 2m from 0̂ to 0̂ in the Hasse diagram of the poset
L(k, j), where 0̂ denotes the unique bottom element (the empty partition) of L(k, j),
the lattice of integer partitions whose shape fits in a k × P
j rectangle, ordered by
2m
inclusion. Let gk,j (m) be this number, and Gk,j (x) =
be the
m gk,j (m)x
generating function.
For j = 1, the number gk,1 (m) counts lattice paths from (0, 0) to (2m, 0) with
steps (1, 1) or (1, −1) that stay between the lines y = 0 and y = k. The evaluation of
gk,1 (m) was first considered by Takács in [28] by a probabilistic argument. Explicit
formula and generating function for this case are well-known. For example, in [19]
one obtains the explicit formula by applying the reflection principle repeatedly, viz.,
X 2m
2m
gk,1 (m) =
−
.
m − i(k + 2)
m + i(k + 2) + k + 1
i
The generating function Gk,1 (x) is a special case of the one for the duration of the
game in the classical ruin problem, that is, restricted random walks with absorbing
barriers at 0 and a, and initial position z. See, for example, Equation (4.11) of
Chapter 14 of [9]: Let
∞
X
Uz (x) =
uz,m xm ,
m=0
18
W. CHEN, E. DENG, R. DU, R. STANLEY, AND C. YAN
where uz,n is the probability that the process ends with the n-th step at the barrier
0. Then
z a−z
q
λ1 (x) − λa−z
(x)
2
Uz (x) =
,
a
a
p
λ1 (x) − λ2 (x)
where
p
p
1 + 1 − 4pqx2
1 − 1 − 4pqx2
λ1 (x) =
,
λ2 (x) =
.
2px
2px
The generating function Gk,1 (x) is just U1 (2x)/x with a = k + 2, z = 1, and
p = q = 1/2.
In general, by the transfer matrix method [24, §4.7]
(5.2)
Gk,j (x) =
det(I − xAk,j (0))
det(I − xAk,j )
is a rational function, where Ak,j is the adjacency matrix of the Hasse diagram
of L(k, j), and Ak,j (0) is obtained from Ak,j by deleting the row and the column
corresponding to 0̂. Ak,j (0) is also the adjacency matrix of the Hasse diagram of
L(k, j) with its bottom element (the empty partition) removed. Note that det(I −
xAk,j ) is a polynomial in x2 since L(k, j) is bipartite [6, Thm. 3.11]. Let det(I −
xAk,j ) = pk,j (x2 ). The following is a table of pk,j (x) for the values of 1 ≤ k ≤ j ≤ 4.
(We only need to list those with k ≤ j since pk,j (x) = pj,k (x).)
√
(k, j) pk,j (x) = det(I − xAk,j )
(1, 1) 1 − x
(1, 2) 1 − 2x
(1, 3) 1 − 3x + x2
(1, 4) (1 − x)(1 − 3x)
(2, 2) (1 − x)(1 − 5x)
(2, 3) (1 − x)(1 − 3x)(1 − 8x + 4x2 ))
(2, 4) (1 − 14x + 49x2 − 49x3 )(1 − 6x + 5x2 − x3 )
(3, 3) (1 − x)(1 − 19x + 83x2 − x3 )(1 − 5x + 6x2 − x3 )2
(3, 4) (1 − 2x)2 (1 − 8x + 8x2 )(1 − 4x + 2x2 )2 (1 − 16x + 60x2 − 32x3 + 4x4 )
·(1 − 24x + 136x2 − 160x3 + 16x4 )
(4, 4) (1 − x)2 (1 − 18x + 81x2 − 81x3 )2 (1 − 27x + 99x2 − 9x3 )(1 − 9x + 18x2 − 9x3 )2
·(1 − 27x + 195x2 − 361x3 )(1 − 6x + 9x2 − x3 )2 (1 − 9x + 6x2 − x3 )2
The polynomial pk,j (x) seems to have a lot of factors. We are grateful to Christian
Krattenthaler for explaining equation (5.4) below, from which we can explain the
factorization of pk,j (x). By an observation [13, §5] of Grabiner, gk,j (n) is equal
to the number of walks with n steps ±ei from (j, j − 1, . . . , 2, 1) to itself in the
chamber j + k + 1 > x1 > x2 > · · · > xj > 0 of the affine Weyl group C̃n . Write
m = j + k + 1. By [13, (23)] there follows
" 2m−1
#j
X
1 X
x2n
= det
sin(πra/m) sin(πrb/m) · exp(2x cos(πr/m))
.
gk,j (n)
(2n)!
m r=0
n
a,b=1
When this determinant is expanded, we obtain a linear combination of terms of the
form
(5.3)
X
xn
exp(2x(cos(πr1 /m)+· · ·+cos(πrj /m))) =
2n (cos(πr1 /m)+· · ·+cos(πrj /m))n ,
n!
n≥0
CROSSINGS AND NESTINGS
19
where 0 ≤ ri ≤ 2m − 1 for 1 ≤ i ≤ j. In fact, the case η = λ of Grabiner’s formula
[13, (23)] shows that the number of walks of length n in the Weyl chamber from
any integral point to itself is again a linear combination of terms 2n (cos(πr1 /m) +
· · · + cos(πrj /m))n . It follows that every eigenvalue of Ak,j has the form
(5.4)
θ = 2(cos(πr1 /m) + · · · + cos(πrj /m)).
(In particular, the Galois group over Q of every irreducible factor of pk,j (x) is
abelian.) Note that a priori not every such θ may be an eigenvalue, since it may
appear with coefficient 0 after the linear combinations are taken. The algebraic
integer z = 2(cos(πr1 /m) + · · · + cos(πrj /m)) lies in the field Q(cos(π/m)), an
extension of Q of degree φ(2m)/2, where φ is the Euler phi-function. To see this,
let z be a primitive 2m-th root of unity. Then z is a root of x + 1/x = 2 cos(π/m).
Hence the field L = Q(z) is quadratic or linear over K = Q(cos(π/m). Since K
is real and L is not for m > 1, we cannot have K = L. Hence [L : K] = 2.
Since [L : Q] = φ(2m), we have [K : Q] = φ(2m)/2. It follows that the minimal
polynomial over Q of z has degree dividing φ(2m)/2. Thus every irreducible factor
of det(I − Ax) has degree dividing φ(2m)/2, explaining why pk,j (x) has many
factors. A more careful analysis should yield more precise information about the
factors of pk,j (x), but we will not attempt such an analysis here.
An interesting special case of determining pk,j (x) is determining its degree, since
the number of eigenvalues of Ak,j equal to 0 is given by j+k
− 2 · deg pk,j (x).
j
Equivalently, since Ak,j is a symmetric matrix, 2 · deg pk,j (x) = rank(Ak,j ). We
have observed the following.
(1) For k + j ≤ 12 and 1 ≤ k ≤ j, Ak,j is invertible exactly for (k, j) =(1, 1),
(1, 3), (1, 5), (1, 7), (1, 9), (1, 11), (3, 3), (3, 7), (3, 9), (5, 5) and (5, 7).
(2) A1,j is invertible if and only if j is odd. This is true because L(1, j) is
a path of length j, whose determinant satisfies the recurrence det(A1,j ) =
− det(A1,j−2 ). The statement follows from the initial conditions det(A1,1 ) =
−1 and det(A1,2 ) = 0.
(3) If Ak,j is invertible, then kj is odd. To see this, let X0 (X1 ) be the set of
integer partitions of even (odd) n whose shape fits in a k × j rectangle.
Since L(k, j) is bipartite graph with vertex partition (X0 , X1 ), a necessary
condition for Ak,j to be invertible is |X0 | = |X1 |. That is, the generat
P
must have a root at q = −1, where
ing function n p(k, j, n)q n = k+j
k
p(k, j, n) is the number of integer partitions on n whose shape fits into a
k × j rectangle. But the multiplicity of 1 + q in the Gaussian polynomial
j
k+j
k
is b k+j
k
2 c − b 2 c − b 2 c, which is 0 unless both j and k are odd.
Item 3 is also proved independently by Jason Burns, who also found a counterexample for the converse: For k = 3 and j = 11, A3,11 is not invertible, in
fact its corank is 6. The invertibility of Ak,j for kj being odd is currently under
investigation.
Acknowledgments
The authors would like to thank Professor Donald Knuth for carefully reading
the manuscript and providing many helpful comments.
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CROSSINGS AND NESTINGS
21
Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P.R. China
E-mail address: chen@nankai.edu.cn
Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P.R. China
Current address: Department of Applied Mathematics, Dalian University of Technology, Dalian,
Liaoning 116024, P.R.China
E-mail address: dengyp@eyou.com
Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P.R. China
Current address: Department of Mathematics, East China Normal University, Shanghai 200062,
P.R. China
E-mail address: rxdu@math.ecnu.edu.cn
Department of Mathematics, Massachusetts Institute of Technology, Cambridge,
MA 02139
E-mail address: rstan@math.mit.edu
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368
E-mail address: cyan@math.tamu.edu